Physical Models of Membrane Behavior Based on the Hodgkin–Huxley Formalism

The electrical behavior of cellular membranes plays a fundamental role in neuronal communication and in many physiological processes involving excitable cells. Mathematical modeling has become an essential tool for understanding the physical mechanisms underlying membrane dynamics and the generation of action potentials. The classical Hodgkin–Huxley model represents the cornerstone of conductance-based descriptions of neuronal activity, providing a quantitative framework in which ionic currents across the membrane are represented through nonlinear differential equations. Over the years, numerous extensions of this model have been developed in order to incorporate additional biophysical mechanisms, including dendritic processing, temperature dependence and electromagnetic effects. However, increasing experimental evidence has shown that neuronal activity is intrinsically stochastic due to the probabilistic nature of ion-channel gating and other microscopic processes. As a consequence, stochastic modeling approaches have been introduced to complement deterministic formulations and to capture the variability observed in real neuronal systems. In this review, we focus on a selected class of membrane models grounded in physical or biophysical principles, namely models that describe membrane dynamics through electrical analogies, conductance-based equations, stochastic channel kinetics, or memory-dependent circuit elements. These approaches can be viewed as extensions, reformulations, or generalizations of the Hodgkin–Huxley framework, developed to address specific physiological or computational limitations. The review focuses on the physical and mathematical structure of selected HH-derived models rather than on their experimental validation, and aims to compare representative physically motivated modeling strategies in terms of their assumptions, interpretability, and domains of applicability.